| L(s) = 1 | + (−0.664 − 1.24i)2-s + (−0.987 − 0.156i)3-s + (−1.11 + 1.65i)4-s + (−1.95 − 1.08i)5-s + (0.461 + 1.33i)6-s + (−2.24 + 2.24i)7-s + (2.81 + 0.291i)8-s + (0.951 + 0.309i)9-s + (−0.0517 + 3.16i)10-s + (5.18 − 1.68i)11-s + (1.36 − 1.46i)12-s + (0.704 + 1.38i)13-s + (4.30 + 1.31i)14-s + (1.76 + 1.37i)15-s + (−1.50 − 3.70i)16-s + (1.01 + 6.39i)17-s + ⋯ |
| L(s) = 1 | + (−0.469 − 0.882i)2-s + (−0.570 − 0.0903i)3-s + (−0.558 + 0.829i)4-s + (−0.874 − 0.484i)5-s + (0.188 + 0.545i)6-s + (−0.849 + 0.849i)7-s + (0.994 + 0.102i)8-s + (0.317 + 0.103i)9-s + (−0.0163 + 0.999i)10-s + (1.56 − 0.507i)11-s + (0.393 − 0.422i)12-s + (0.195 + 0.383i)13-s + (1.14 + 0.350i)14-s + (0.455 + 0.355i)15-s + (−0.376 − 0.926i)16-s + (0.245 + 1.55i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.586617 + 0.0550679i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.586617 + 0.0550679i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.664 + 1.24i)T \) |
| 3 | \( 1 + (0.987 + 0.156i)T \) |
| 5 | \( 1 + (1.95 + 1.08i)T \) |
| good | 7 | \( 1 + (2.24 - 2.24i)T - 7iT^{2} \) |
| 11 | \( 1 + (-5.18 + 1.68i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.704 - 1.38i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-1.01 - 6.39i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-1.67 - 1.21i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.0339 - 0.0666i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (-0.915 - 1.26i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (0.382 - 0.525i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (6.97 - 3.55i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (0.243 - 0.750i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-8.59 - 8.59i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.848 + 5.35i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-1.48 + 9.36i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (2.30 - 7.08i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.12 - 6.52i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (1.91 - 0.304i)T + (63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (-1.60 - 2.20i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.35 + 1.20i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (7.91 - 5.75i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.895 - 5.65i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (17.4 - 5.68i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.17 - 0.343i)T + (92.2 + 29.9i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.82983952935247122908096871998, −11.07366612354513834189908301496, −9.905026398551938210863560150159, −8.944005023163300533611570394124, −8.364946584407233362682301004597, −6.94193802135815312410680908715, −5.81260235854173052449662620162, −4.22765531060325164680138212990, −3.39040843479327790622347511910, −1.34800439304747076289178924384,
0.64152953991602141706051312140, 3.62827861468274918720455873115, 4.63435202953935067231016371581, 6.08997881744254479703834395647, 7.12495134705196469823510373280, 7.32244711196911628270181632865, 8.950192667523354022300424821894, 9.759616934808480664811258374831, 10.64035769470584924874079918590, 11.58272279393789906487317521408
